For our further studies, box-visibility representations of graphs
and the number of crossings will be useful.
In a box-visibility representation, the
vertices of a graph are represented as boxes, which are axis
aligned rectangles. Each edge is an orthogonal polyline with
horizontal and vertical segments between the borders of two boxes,
so that segments do not traverse other boxes. Edges incident to the
same box do not cross. There is a rectangle visibility
representation [1]} if each edge consists of a
horizontal or a vertical segment, and a special box-visibility
representation if each edge consists of a vertical and a horizontal
segment with a bend to the right in between. It is well-known that
every graph has a special box-visibility representation. The idea
dates back to the early 1980th and Valiant's representation of
graphs of degree at most four [2]}. Simply, place the
boxes for the vertices on the main diagonal and route each edge
\(u v\)  by a vertical segment from the top of the box of \(u\) 
and a horizontal segment to the left side of the box of \(v\) . Later
studies have focussed on box-visibility representation with small
area and few bends [3]}, [4]}, [5]}.
Box-visibility representations have been used to show that every
graph has a 3-bend RAC drawing [6]}. There is an
alternative 3-bend RAC drawing, in which the vertices are placed on
a horizontal line and the second and third segments are drawn with
slope \(\pm 1\) .
